Parallel and totally umbilical hypersurfaces of the four‐dimensional Thurston geometry Sol04$\text{Sol}^4_0$

Marie D'haene, Jun‐ichi Inoguchi and Joeri Van der Veken

Mathematische Nachrichten, 2024, vol. 297, issue 5, 1879-1891

Abstract: We study hypersurfaces of the four‐dimensional Thurston geometry Sol04$\mathrm{Sol}^4_0$, which is a Riemannian homogeneous space and a solvable Lie group. In particular, we give a full classification of hypersurfaces whose second fundamental form is a Codazzi tensor—including totally geodesic hypersurfaces and hypersurfaces with parallel second fundamental form—and of totally umbilical hypersurfaces of Sol04$\mathrm{Sol}^4_0$. We also give a closed expression for the Riemann curvature tensor of Sol04$\mathrm{Sol}^4_0$, using two integrable complex structures.

Date: 2024
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https://doi.org/10.1002/mana.202300372

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